4T Sustainnability Microrandomizad Trial
Power simulations
Model
\[ TIR_{ij} = \beta_0 + \gamma_i + \beta_1 Trt_{ij} + \beta_2 Time_{ij} + \beta_3 Trt_{ij} Time_j + e_{ij}, \]
with
\[ \gamma_i \sim N(0, \sigma_\gamma^2), \quad e_{ij} \sim N(0, \sigma_e^2), \quad \gamma_i \perp e_{ij} \]
Also, \(\beta_1 = 0\) since, at baseline, the treatment effect is zero. The correlation between two TIRs for the same patient will be given by
\[ \rho = Cor(TIR_{ij}, TIR_{ij'}) = (1 + (\sigma_e/\sigma_\gamma)^2)^{-1}. \]
In all simulated scenarios, we fixed \(\sigma_e/\sigma_\gamma = 0.5\), such that \(\rho = 0.8\). Also, we assume a baseline TIR before microrandomization of \(\beta_0 = 0.75\) and a baseline time trend of \(\beta_2 = -0.005\) as patients get progressively worse TIR if not intervened.
Thus, using sample sizes of 50, 100, and 200, we simulate
\[ TIR_{ij} = .75 + \gamma_i - 0.005 Time_{ij} + \beta_3 Trt_{ij} Time_j + e_{ij}, \]
under the following scenarios
Scenarios A1-A5
- \(\beta_1 = 0.005, 0.0025, 0.001, 0.0005, 0.0001\)
- \(\sigma_e = 0.1\) and \(\sigma_\gamma = 0.2\)
Scenarios B1-B5
- \(\beta_1 = 0.005, 0.0025, 0.001, 0.0005, 0.0001\)
- \(\sigma_e = 0.08\) and \(\sigma_\gamma = 0.16\)
Scenarios C1-C2
- \(\beta_1 = 0\)
- C1: \(\sigma_e = 0.1\) and \(\sigma_\gamma = 0.2\)
- C2: \(\sigma_e = 0.08\) and \(\sigma_\gamma = 0.16\)
Figure 1 has data from a non-microrandomized sample of size 100, i.e., patients simply randomized to default or add-on treatment at baseline.
Simulation results
The results below were obtained using \(B = 2000\) replications of the scenarios described above.
